arXiv subjects

Clemens Kirisits

Publications and source records attributed to Clemens Kirisits.

18 recordsLinked to original sources

arXiv2025

Regularization of Nonlinear Inverse Problems -- From Functional Analysis to Data-Driven Approaches

The focus of this book is on the analysis of regularization methods for solving \emph{nonlinear inverse problems}. Specifically, we place a strong emphasis on techniques that incorporate supervised or unsupervised data derived from prior experiments. This approach enables the integration of data-driven insights into the solution of inverse problems governed by physical models. \emph{Inverse problems}, in general, aim to uncover the \emph{inner mechanisms} of an observed system based on indirect or incomplete measurements. This field has far-reaching applications across various disciplines, such as medical or geophysical imaging, as well as, more broadly speaking, industrial processes where identifying hidden parameters is essential.

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arXiv2024

Generalized Fourier Diffraction Theorem and Filtered Backpropagation for Tomographic Reconstruction

This paper concerns diffraction-tomographic reconstruction of an object characterized by its scattering potential. We establish a rigorous generalization of the Fourier diffraction theorem in arbitrary dimension, giving a precise relation in the Fourier domain between measurements of the scattered wave and reconstructions of the scattering potential. With this theorem at hand, Fourier coverages for different experimental setups are investigated taking into account parameters such as object orientation, direction of incidence and frequency of illumination. Allowing for simultaneous and discontinuous variation of these parameters, a general filtered backpropagation formula is derived resulting in an explicit approximation of the scattering potential for a large class of experimental setups.

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arXiv2024

Using the Transport of Intensity and the Transport of Phase Equation for Phase Retrieval

We investigate the transport of intensity equation (TIE) and the transport of phase equation (TPE) for solving the phase retrieval problem. Both the TIE and the TPE are derived from the paraxial Helmholtz equation and relate phase information to the intensity. The TIE is usually favored since the TPE is nonlinear. The main contribution of this paper is a discussion of preferential use of either one of the two equations or potential benefits of a hybrid use. Moreover, we discuss the solution of the TPE with the method of characteristics and with viscosity methods. Both the TIE and the viscosity method are numerically implemented with finite element methods.

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arXiv2024

Diffraction Tomography for a Generalized Incident Field

Diffraction tomography is an inverse scattering technique used to reconstruct the spatial distribution of the material properties of a weakly scattering object. The object is exposed to radiation, typically light or ultrasound, and the scattered waves induced from different incident field angles are recorded. In conventional diffraction tomography, the incident wave is assumed to be a monochromatic plane wave, an unrealistic simplification in practical imaging scenarios. In this article, we extend conventional diffraction tomography by introducing the concept of customized illumination scenarios, with a pronounced emphasis on imaging with focused beams. We present a new forward model that incorporates a generalized incident field and extends the classical Fourier diffraction theorem to the use of this incident field. This yields a new two-step reconstruction process which we comprehensively evaluate through numerical experiments.

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arXiv2023

Uncertainty Quantification for Scale-Space Blob Detection

We consider the problem of blob detection for uncertain images, such as images that have to be inferred from noisy measurements. Extending recent work motivated by astronomical applications, we propose an approach that represents the uncertainty in the position and size of a blob by a region in a three-dimensional scale space. Motivated by classic tube methods such as the taut-string algorithm, these regions are obtained from level sets of the minimizer of a total variation functional within a high-dimensional tube. The resulting non-smooth optimization problem is challenging to solve, and we compare various numerical approaches for its solution and relate them to the literature on constrained total variation denoising. Finally, the proposed methodology is illustrated on numerical experiments for deconvolution and models related to astrophysics, where it is demonstrated that it allows to represent the uncertainty in the detected blobs in a precise and physically interpretable way.

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arXiv2021

Diffraction Tomography, Fourier Reconstruction, and Full Waveform Inversion

In this paper, we study the mathematical imaging problem of diffraction tomography (DT), which is an inverse scattering technique used to find material properties of an object by illuminating it with probing waves and recording the scattered waves. Conventional DT relies on the Fourier diffraction theorem, which is applicable under the condition of weak scattering. However, if the object has high contrasts or is too large compared to the wavelength, it tends to produce multiple scattering, which complicates the reconstruction. We give a survey on diffraction tomography and compare the reconstruction of low and high contrast objects. We also implement and compare the reconstruction using the full waveform inversion method which, contrary to the Born and Rytov approximations, works with the total field and is more robust to multiple scattering.

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arXiv2021

Fourier reconstruction for diffraction tomography of an object rotated into arbitrary orientations

In this paper, we study the mathematical imaging problem of optical diffraction tomography (ODT) for the scenario of a microscopic rigid particle rotating in a trap created, for instance, by acoustic or optical forces. Under the influence of the inhomogeneous forces the particle carries out a time-dependent smooth, but complicated motion described by a set of affine transformations. The rotation of the particle enables one to record optical images from a wide range of angles, which largely eliminates the "missing cone problem" in optics. This advantage, however, comes at the price that the rotation axis in this scenario is not fixed, but continuously undergoes some variations, and that the rotation angles are not equally spaced, which is in contrast to standard tomographic reconstruction assumptions. In the present work, we assume that the time-dependent motion parameters are known, and that the particle's scattering potential is compatible with making the first order Born or Rytov approximation. We prove a Fourier diffraction theorem and derive novel backprojection formulae for the reconstruction of the scattering potential, which depends on the refractive index distribution inside the object, taking its complicated motion into account. This provides the basis for solving the ODT problem with an efficient non-uniform discrete Fourier transform.

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arXiv2019

Preservation of Piecewise Constancy under TV Regularization with Rectilinear Anisotropy

A recent result by Lasica, Moll and Mucha about the $\ell^1$-anisotropic Rudin-Osher-Fatemi model in $\mathbb{R}^2$ asserts that the solution is piecewise constant on a rectilinear grid, if the datum is. By means of a new proof we extend this result to $\mathbb{R}^n$. The core of our proof consists in showing that averaging operators associated to certain rectilinear grids map subgradients of the $\ell^1$-anisotropic total variation seminorm to subgradients.

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arXiv2019

Invariant $φ$-minimal sets and total variation denoising on graphs

Total variation flow, total variation regularization and the taut string algorithm are known to be equivalent filters for one-dimensional discrete signals. In addition, the filtered signal simultaneously minimizes a large number of convex functionals in a certain neighbourhood of the data. In this article we study the question to what extent this situation remains true in a more general setting, namely for data given on the vertices of a finite oriented graph and the total variation being $J(f) = \sum_{i,j} |f(v_i) - f(v_j)|$. Relying on recent results on invariant $φ$-minimal sets we prove that the minimizer to the corresponding Rudin-Osher-Fatemi (ROF) model on the graph has the same universal minimality property as in the one-dimensional setting. Interestingly, this property is lost, if $J$ is replaced by the discrete isotropic total variation. Next, we relate the ROF minimizer to the solution of the gradient flow for $J$. It turns out that, in contrast to the one-dimensional setting, these two problems are not equivalent in general, but conditions for equivalence are available.

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arXiv2019

On the ROF Model in Rectilinear Anisotropy: Piecewise Constant Approximation and Universal Minimality

We prove that the $L^2$ distance between the minimizer of the $\ell^1$-anisotropic Rudin-Osher-Fatemi (ROF) functional and its minimizer over the space of piecewise constant functions on a rectilinear grid is $\mathcal{O}(h^{\frac12 - \frac{q'}{2q}})$, where $h$ is the grid's mesh size and the datum belongs to $L^q$, $q \ge 2$. These convergence rates are valid in any dimension $d\ge 1$. However, in dimension $d = 1$ they can be further improved to $\mathcal{O}(h^{\frac12 - \frac{1}{2q}})$. To establish the error bounds, $L^q$ estimates of the ROF minimizer in terms of the datum are critical. Such estimates are particular cases of a universal minimality property of the ROF minimizer derived in the second part of the paper. There it is shown, in both the finite-dimensional and infinite-dimensional settings, that the minimizer simultaneously minimizes a broad class of convex functionals over a neighbourhood of the datum arising in the convex dual of the ROF problem. This extends previous results of similar type about taut strings and the ROF problem.

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arXiv2018

A range condition for polyconvex variational regularization

In the context of convex variational regularization it is a known result that, under suitable differentiability assumptions, source conditions in the form of variational inequalities imply range conditions, while the converse implication only holds under an additional restriction on the operator. In this article we prove the analogous result for polyconvex regularization. More precisely, we show that the variational inequality derived by Kirisits, Scherzer (2017) implies that the derivative of the regularization functional must lie in the range of the dual-adjoint of the derivative of the operator. In addition, we show how to adapt the restriction on the operator in order to obtain the converse implication.

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arXiv2016

Convergence rates for regularization functionals with polyconvex integrands

Convergence rates results for variational regularization methods typically assume the regularization functional to be convex. While this assumption is natural for scalar-valued functions, it can be unnecessarily strong for vector-valued ones. In this paper we focus on regularization functionals with polyconvex integrands. Even though such functionals are nonconvex in general, it is possible to derive linear convergence rates with respect to a generalized Bregman distance, an idea introduced by Grasmair in 2010. As a case example we consider the image registration problem.

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arXiv2015

Convective regularization for optical flow

We argue that the time derivative in a fixed coordinate frame may not be the most appropriate measure of time regularity of an optical flow field. Instead, for a given velocity field $v$ we consider the convective acceleration $v_t + \nabla v v$ which describes the acceleration of objects moving according to $v$. Consequently we investigate the suitability of the nonconvex functional $\|v_t + \nabla v v\|^2_{L^2}$ as a regularization term for optical flow. We demonstrate that this term acts as both a spatial and a temporal regularizer and has an intrinsic edge-preserving property. We incorporate it into a contrast invariant and time-regularized variant of the Horn-Schunck functional, prove existence of minimizers and verify experimentally that it addresses some of the problems of basic quadratic models. For the minimization we use an iterative scheme that approximates the original nonlinear problem with a sequence of linear ones. We believe that the convective acceleration may be gainfully introduced in a variety of optical flow models.

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arXiv2014

Optical Flow on Evolving Surfaces with Space and Time Regularisation

We extend the concept of optical flow with spatiotemporal regularisation to a dynamic non-Euclidean setting. Optical flow is traditionally computed from a sequence of flat images. The purpose of this paper is to introduce variational motion estimation for images that are defined on an evolving surface. Volumetric microscopy images depicting a live zebrafish embryo serve as both biological motivation and test data.

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arXiv2014

Optical Flow on Moving Manifolds

Optical flow is a powerful tool for the study and analysis of motion in a sequence of images. In this article we study a Horn-Schunck type spatio-temporal regularization functional for image sequences that have a non-Euclidean, time varying image domain. To that end we construct a Riemannian metric that describes the deformation and structure of this evolving surface. The resulting functional can be seen as natural geometric generalization of previous work by Weickert and Schn\"orr (2001) and Lef\`evre and Baillet (2008) for static image domains. In this work we show the existence and wellposedness of the corresponding optical flow problem and derive necessary and sufficient optimality conditions. We demonstrate the functionality of our approach in a series of experiments using both synthetic and real data.

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arXiv2014

Decomposition of Optical Flow on the Sphere

We propose a number of variational regularisation methods for the estimation and decomposition of motion fields on the $2$-sphere. While motion estimation is based on the optical flow equation, the presented decomposition models are motivated by recent trends in image analysis. In particular we treat $u+v$ decomposition as well as hierarchical decomposition. Helmholtz decomposition of motion fields is obtained as a natural by-product of the chosen numerical method based on vector spherical harmonics. All models are tested on time-lapse microscopy data depicting fluorescently labelled endodermal cells of a zebrafish embryo.

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arXiv2013

Optical Flow on Evolving Surfaces with an Application to the Analysis of 4D Microscopy Data

We extend the concept of optical flow to a dynamic non-Euclidean setting. Optical flow is traditionally computed from a sequence of flat images. It is the purpose of this paper to introduce variational motion estimation for images that are defined on an evolving surface. Volumetric microscopy images depicting a live zebrafish embryo serve as both biological motivation and test data.

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arXiv2012

Total Variation Denoising on Hexagonal Grids

This work combines three paradigms of image processing: i) the total variation approach to denoising, ii) the superior structure of hexagonal lattices, and iii) fast and exact graph cut optimization techniques. Although isotropic in theory, numerical implementations of the $BV$ seminorm invariably show anisotropic behaviour. Discretization of the image domain into a hexagonal grid seems perfectly suitable to mitigate this undesirable effect. To this end, we recast the continuous problem as a finite-dimensional one on an arbitrary lattice, before focussing on the comparison of Cartesian and hexagonal structures. Minimization is performed with well-established graph cut algorithms, which are easily adapted to new spatial discretizations. Apart from producing minimizers that are closer in the $\ell^1$ sense to the clean image for sufficiently high degrees of regularization, our experiments suggest that the hexagonal lattice also allows for a more effective reduction of two major drawbacks of existing techniques: metrication artefacts and staircasing. For the sake of practical relevance we address the difficulties that naturally arise when dealing with non-standard images.

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